Valuation of Convertible Bonds pdf

Sanaa Khan K1306336
FACULTY OF SCIENCE,
ENGINEERING AND COMPUTING
School of Computer Science and
Mathematics
BSc (Hons) DEGREE
IN
Financial Mathematics with Business Management
Name: Sanaa Khan
ID Number: K1306336
Project Title: Valuation of Convertible Bonds
Date: 11/04/16
Supervisor: Luluwah Al-Fagih
WARRANTY STATEMENT
This is a student project. Therefore, neither the student nor Kingston University
makes any warranty, express or implied, as to the accuracy of the data or conclusion
of the work performed in the project and will not be held responsible for any
consequences arising out of any inaccuracies or omissions therein.

Abstract
In this paper, we will be discussing methods of pricing a European style convertible bond
(CB), i.e. where conversion can only take place at maturity. Pricing methods include using
the Black-Scholes model to price the bond by splitting components to help simplify the
procedure. Furthermore, contract features will be looked upon, to give a better perspective
as to what is said between the issuer and the bondholder, as well as how the CB is formed
and the features within it. The paper will also be looking at the analysis of price sensitivities
and how different features affect the price of a CB and the impact they have on a portfolio
containing a CB.

Sanaa Khan K1306336
Contents
Abstract i
1. Introduction 1
2. Payoff Profiles 4
2.1. Notation 4
2.2. Bondholder’s Perspective 5
2.3. Bond Issuer’s Perspective 5
2.4. Payoff 6
2.5. A Zero-Sum Game 7
2.5.1. Example 7
3. Contract Features of a Convertible Bond 9
3.1. Convertible Bond Financing 9
3.2. Maturity 9
3.3. Principle 9
3.4. Conversion Ratio 10
3.5. Call Provisions 10
3.6. Put Provisions 10
3.7. Coupon Payments 11
3.8. Refix Clause 11
3.9. Other Non-Standard Clauses 12
3.10. Termination 12

Sanaa Khan K1306336
4. Properties of a Convertible Bond 13
4.1. Conversion Price 13
4.2. Parity 13
4.3. Premium to Parity 14
4.4. Investment Premium 14
4.5. Bond Floor 15
4.6. Price Sensitivities 16
4.7. Upper and Lower Bounds 19
5. Pricing Methods 21
5.1. Mathematical Background 21
5.2. Monte Carlo Simulation 21
5.3. Lattice based Method 22
5.4. Reduced Form Approach 22
5.5. Tsiveriotis-Fernandes Method 23
5.6. Black-Scholes Method 23
6. Black-Scholes Model 26
6.1. The Bond Price 26
6.2. Example 29
6.3. Margrabe Formula 31
7. Conclusion 34
8. References 35

1
1. Introduction
Convertible bonds (CB) were first used during the 1960s. Convertible bonds are hybrid
securities; they use both equity and debt. A convertible bond is a bond such that the holder
of the bond; that being the investor is able to convert it into cash or equity when they feel it
would be beneficial to them [7 - pg 58]. Ingersoll’s (1977) research suggests that the general
valuation procedure would be to set up the price of the convertible and equate it to the
maximum value of a straight bond, or the value it holds within the common stock (after
conversion) given that at some point in the near future. The value found from this, would
then be discounted back to the present value. Yan, Yi, Yang and Liang (2015) state they wish
to keep hold of the bond, in which case they will receive interest payments; or they could
convert it into the company’s stocks. The bondholder would ideally pick a strategy in which
they would be able to maximise the CB value.
The issuers of convertible bonds are usually smaller firms. Smaller firms who are looking into
getting finances. The reason for this is because smaller firms are not as well-known and need
financing when their credit is low [20]. It is found that when a weaker firm wishes to issue a
CB, it shows they have faith in their project. This enhances their chances of gaining investors
for their company. However, a larger firm would not need to issue convertible bonds as they
would easily be able to get funding and or loans as they are more known within the industry.
If a larger firm wanted to issue a bond, they would not have enough buyers.
The motivation behind the smaller firms issuing the convertible bonds is due to the fact they
lack stable credit histories. This means they would have to pay higher interest payments;
also known as coupons - to their debt holders. The size of a firm usually is a reason as to why
there is an issuance of convertible bonds [12]. Firm size is associated with bankruptcy costs;
since smaller firms are more vulnerable to failure and are risk averse. Smaller firms face

Sanaa Khan K1306336
2
higher degree information asymmetry, this could increase the cost of the debt. It could also
lead to having more restrictive contracts – also known as covenants, had they wanted to
issue a straight bond. This is a reason why larger firms just offer straight bonds. A convertible
bond is more flexible the way it works, matters are stated within the contract, as well as
being set out if the firm breaks the contract they (the bondholder) will receive a premium
[12]. The motivation behind issuing CBs is the fact firms will have interest rate-cost savings,
in comparison to issuing straight corporate1
bonds [20].
Another reason why firms issue CBs is to ensure the investor has no entitlement in the
running of the business. This would mean having the ability to vote for the directors that
would only be in control of the common stockholders. This makes it attractive to firms, as
they know their positions will not be endangered nor questioned. Kwok (2014) suggests that
convertible bonds are chosen by firms over straight bonds due to the lower coupon rate.
CBs have a callable feature which means it can be redeemed by the issuer prior to the
contractual date, this paper will follow a European styled CB. At this point, a price – in the
form of a penalty, would be paid to the bondholder, as the company is forcing them to
either convert or surrender the bond [ref 7 page 58].
Owning a convertible bond is like playing a game. The bondholder is allowed to convert the
bond when they see it is beneficial for them. Suppose the bondholder converts before the
call date set within the covenant; it would mean the shareholders were not able to call the
bond when they thought it would be beneficial for them [7]. According to Yan, Yi, Yang and
Liang (2015) when the coupon rate is bounded above by the interest rate multiplied by the
strike price, that is when the bondholder will convert the CB. The conversion for the issuer
will take place when the coupon rate is lower than the dividend rate multiplied by the strike
1
Information Asymmetry: a party within a transaction has more information on the other party that
they are dealing with. Due to this, a party is likely to take advantage of the other party’s lack of
knowledge.

Sanaa Khan K1306336
3
price; though this paper will not be discussing dividends used within CBs. The contract is
terminated when the coupon rate lies in between the two bounds, at that point both parties
will terminate the contract.
The bondholder will be receiving coupon payments, over the life of the CB, up until the
contract has reached its expiry (maturity). Prior to maturity, the bondholder has the right to
convert their bond into the company’s shares. Close to the end of the contract the company
have the right to call the bond back and force the bondholder to capitulate the bond to the
company.
In this paper, we will be discussing methods of pricing a European style convertible bond
(CB), i.e. where conversion can only take place at maturity. Pricing methods include using
the Black-Scholes model to price the bond by splitting components to help simplify the
procedure. Furthermore, contract features will be looked upon, to give a better perspective
as to what is said between the issuer and the bondholder, as well as how the CB is formed
and the features within it. The paper will also be looking at the analysis of price sensitivities
and how different features affect the price of a CB and the impact they have on a portfolio
containing a CB.

Sanaa Khan K1306336
4
2. Payoff Profiles
A payoff is what is received by the bondholder during the lifetime of the bond. The
bondholder has two options, (i) to receive the face value, or (ii) the share price multiplied by
the conversion ratio – the one with the greater value is what the bondholder will receive.
First we introduce some notation that will be used throughout the paper.
2.1 Notation
• 𝑁 – Face value
• 𝐶𝑟 – Conversion ratio
• 𝑆𝑡 – Share price at time 𝑡
• 𝐶 – Coupon payment
• 𝑇 – Maturity
• 𝐶 𝑃 – Conversion price
• 𝐾 – Strike price
• 𝐵 𝐹 – Bond floor
• 𝑃 – Price of CB. 𝑃 = 𝑃(𝑡) – price of CB at time 𝑡
• 𝑃𝑎 – Parity
• 𝑟 – interest rate
• 𝑐𝑡 – European call option

Sanaa Khan K1306336
5
2.2 Bondholder’s Perspective:
The bondholder’s perspective is very important during the issuance of the bond, all the way
up until maturity. In order to show their perspective the diagram below shows the true
picture of what they expect in return.
The four scenarios show what the payoff would be when the bond is converted, the
bondholder would receive their payoff; however, they would receive the maximum out of
the two. The next is the issuer calling the bond earlier, that way they end up paying them
the exercise price. When the bond is not exercised from either side the face value is given to
the bondholder. The default value would be received when the value is below the bond
floor; knowing the bond floor is the lowest boundary, below that the bondholder would then
receive zero.
2.3 Bond Issuer’s Perspective:
When the bondholder converts, the issuer pays the bondholder the maximum value out of
the ⁡( 𝑆𝑡 𝐶𝑟, 𝑁). When the issuer voluntarily calls the bond they pay the strike to the
bondholder. When the bond is not exercised by either party the face value is paid back to
Conversion: max⁡( 𝑆𝑡 𝐶𝑟, 𝑁)
Not exercised: 𝑁
Default: 0
Issuer calls bond: 𝐾

Sanaa Khan K1306336
6
the holder at maturity. Default takes place when the value of the CB has fallen below the
bond floor.
Both parties have the opportunity to gain and to lose, but the total of the sum of their
choices will equal to zero. Following these scenarios we can see, how a zero sum game is
easily associated with CBs, regardless of the decision or the option which is put in front of
the two parties (See section 2.5.1 below).
2.4 Payoff
The payoff is a function [26]: Payoff:⁡𝑚𝑎𝑥(⁡𝑁, 𝐶𝑟 𝑆 𝑇); that tells the bondholder how much
they will receive from the bond issuer. When the bondholder decides to convert the bond
they will be receiving the maximum amount out of the two values. This means that the share
price is low the bondholder will not be converting and would prefer to receive the face value
𝑁. However, they will receive amount 𝐶𝑟 𝑆 𝑇 when the share price is high and the bondholder
chooses to convert. However, if there is a final coupon payment which is to be made then
the function changes to: 𝑚𝑎𝑥(𝑁 + ⁡𝐶, 𝐶𝑟 𝑆 𝑇).⁡Again, the same rule applies; whichever value
is higher is what the bondholder will receive at conversion.
Therefore, the value of the CB at maturity is given by:
𝑃(𝑇) = 𝐶𝑟 𝑆 𝑇⁡⁡⁡⁡⁡⁡𝑖𝑓⁡𝐶𝑟 𝑆 𝑇 ≥ 𝑁
𝑃(𝑇) = 𝑁 + 𝐶⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓⁡𝐶𝑟 𝑆 𝑇 < 𝑁 + 𝐶

Sanaa Khan K1306336
7
2.5 A Zero-sum Game
A zero-sum game is a game where if one player gains then the other makes a loss. A CB can
be seen as a zero-sum game since the payoff of the CB follows the same ideology. Only one
party will be gaining something out of the game and one will lose, making the sum of the
game equal to zero. Each player uses a strategy to ensure that they are reducing their
opponent’s payoff.
Nash equilibrium (NE) is a term used within the theory of games; it is the solution concept of
a competitive game between two or more players. It is a way in which strategies are used to
make a profit. The game consists of: a set of actions and the choices of the set actions and
the impact they have on each player. The participants of the game are known to be in NE
when making strategic decisions, whilst considering their opponents decisions too. It is seen
that NE does not mean there will be a larger payoff amount necessarily for all players within
the game; it could be the case where a player receives a smaller amount due to the choice
they make [9].
2.5.1 Example of a zero-sum game:
To illustrate this in more detail, we look at a general example:
Two players within the game (the investor and the issuer) are both playing for the higher
payoff. The first player (purple) picks one of the two actions, either 1 or 2, without sharing it
with the other party. The second player (green) then picks an option out of the three
available choices. Again, player 2 chooses without player 1 knowing their decision. This then
leads to their choices being revealed, and the players are able to see their points and the
impact it has had on the payoff due to their choice. In this case, if purple picked option 2 and

Sanaa Khan K1306336
8
green decided to pick B, it would come to be that purple has gained 20 points and green has
now lost 20 points.
𝑨 𝑩 𝑪
𝟏 30, ⁡ − 30 −10, ⁡10 20, ⁡ − 20
𝟐 10, ⁡ − 10 20, ⁡ − 20 −20, ⁡20
The next step they would be taking is to ensure they are able to maximise their payoff.
Purple in this case could then say “With the second option, I could lose 20 and only win 20,
but with option 1 I could lose 10 but gain 30, which mean option 1 is more beneficial.”
Having the same strategy, Green would pick option C, that way they could gain 20, only if
Purple has picked option 2. On the other hand, if Purple were to pick option 2, only to know
that Green is more likely to pick option B. The strategy behind this would mean that Green
would then have already chosen option B, and player Purple would pick option 2; which
would lead to Purple gaining 20 and Green losing 20. Both players would be playing in order
to gain the highest payoff possible, that too with the intention of knowing what the other
player has chosen.
The probability within this example reveals that Purple should choose option 1, which has
the probability of 4/7 and option 2 which has the probability of 3/7. Whereas Green should
set the probabilities: 0, 4/7 and 3/7 to the three options A, B and C. In this case Purple will
then be gaining 20/7 on an average per game the two players are involved in.

Sanaa Khan K1306336
9
3. Contract features of a Convertible Bond (CB)
In this section, we look at the contract features in a CB.
3.1. CB Financing
A feature within a covenant is CB financing, this is the feature which meets the needs of the
issuer and the borrower [11].
3.2. Maturity: 𝑻
Maturity is the end of the contractual date set for the bond, also known as the expiration
date. This is when the firm has to pay back the entire amount back to the investor. Chan and
Chen (2004) state that maturity is usually between the years “…2, 3, 5, 7 and 10…” [6-pg 6]
however, it is possible to have some which last longer than 10 years. Brennan and Schwartz
(1980) found that the value of the CB depends on the maturity, as it has an impact on the
underlying asset risk of the company issuing the bond.
3.3. Principal or Face Value: 𝑵
A principal is the face value of the CB, the amount that the bond can be redeemed at
maturity. There have been occasions where the bond has been redeemed at maturity for a
larger contracted price, than the principal of the bond [11]. This is usually due to the change
in share price, when there is an increase that is when the investor receives a larger amount
at conversion.

Sanaa Khan K1306336
10
3.4. Conversion Ratio: 𝑪 𝒓
Conversion ratio is mentioned within the covenant, it is highly common for there to be a
scheduled timing as to when the conversion of the ratio takes place. It is usually adjusted
over the life of the CB. The conversion ratio determines how many shares the bondholder
will receive at conversion. The way the conversion ratio works is the par value of the
convertible bond, would be divided by conversion price, which would then all be multiplied
by the price per share. This would be a method in which the value would be evaluated, as
well as the firm knowing how much they would have to pay [11]. An example to follow this
would be: If the company has the par value set to £1000 and the conversion ratio has been
set to 25 shares, using this information we can find that the conversion price would be £40.
3.5. Call Provisions: 𝒄𝒕
CBs tend to have a call feature, this allows the investor to know the CB would be called back.
This is when the firm decides to purchase the CB back at a particular date and time. The firm
can force the investor to convert/surrender the bond to them within a brief period of time
[11]. When a bond is called before maturity, the firm pay a penalty, which is pre-set. Lau and
Kwok (2004) suggest that the firm should call back the bond when it reaches the call price.
3.6. Put Provisions: 𝒑 𝒕
Put provisions is a least common factor used within the contract. This is when the
bondholder is able to sell the CB back to the firm at a particular price and date [11]. Usually,
the firm will set a date when the put provision can start, as the contract would contain a
statement which states that the bondholder would have to keep the bond for a certain

Sanaa Khan K1306336
11
period of time before they convert the bond. However, a put provision has specified dates; it
is not continued throughout the life of the CB. A put provision is in place to increase the
protection of the bondholder, which leads to the increase in value of the CB [30].
3.7. Coupon Payments: 𝑪
Coupon payments are for the investor to gain over the period of the bond. It is agreed upon
how often they are to be received, i.e. monthly, annually or half-yearly. Amiram, Kalay, Kalay
and Ozel (2014) have suggested firms that face a higher information asymmetry are more
likely to issue bonds with a higher coupon value. An influential factor for coupon payments is
agency conflicts. This is when the shareholder wishes to gain an increase in the share value,
but the management are not cooperating. The coupon payment is a contractual term which
is able to reduce the agency conflicts. Lastly, the firms which face intense agency conflicts
tend to issue CBs with higher coupon rates [7].
3.8. Refix Clause
A refix clause is a feature which is used in the contract to make it more desirable to the
investor. It alternates the conversion ratio or the conversion price, which is subjected to
share price level between the issuance of the bond, up until maturity. The refix clause adds
additional value to the investor, which means an increase in the premium price paid for the
CB. The refix ensures the bondholder is protected against the decrease in share price [11].

Sanaa Khan K1306336
12
3.9. Other Non-Standard Clauses
The conversion segment usually states that the bondholder will receive a combination of
assets, both shares and cash. It is not necessary that they receive just shares when they
convert the CB [11].
3.10. Termination
Termination is when the contract is cancelled, which means the investor will no longer be
receiving any coupon payments. There are three occasions where the contract is terminated:
if the issuer calls back the bond before maturity, if the bondholder chooses to convert any
time up until maturity or if both the issuer and the bondholder choose to stop altogether.
The second scenario will mean the investor will be receiving 𝐶𝑟 𝑆𝑡 at time 𝑡, with the
prearranged conversion rate 𝐶𝑟. The last scenario is if neither party exercises the CB from
the issuance up until the maturity. The bondholder is then expected to sell the CB back to
the issuer at maturity. However, when the bondholder sells the CB back, it is expected to be
sold according to the pre-set amount or the other option is to convert it into equity at the
conversion rate 𝐶𝑟 [29].

Sanaa Khan K1306336
13
4. Properties of a Convertible Bond (CB)
In this section we will be looking at the properties of a CB and how they are derived and
simplified.
4.1 Conversion Price: CP
The conversion price (𝐶 𝑃) is the price per share at which the CB can be converted to stock.
The strike of each call option is equal to the conversion price: 𝐶 𝑃. So, 𝐾⁡ =⁡ 𝐶 𝑃. The formula
shows how the 𝐶 𝑃 is found for the bond. The formula for the conversion price is equal to the
face value divided by the conversion ratio [26].
4.2 Parity: Pa
The parity is the value of the shares that a party would receive if the CB were to be
converted immediately. The parity is usually a percentage of the face value of the CB. It uses
the share price which is multiplied by the conversion ratio, which is then divided by the face
value. The value of 𝐶 𝑃 is the reciprocal of the face value and the conversion ratio, which is
why it is easily substituted within the formula for the parity. When the face value is at par or
1, then the parity is equal to the share price multiplied by the conversion ratio [26].
𝑃𝑎 =⁡
𝑆⁡ × 𝐶𝑟
𝑁

Sanaa Khan K1306336
14
=
𝑆
𝐶 𝑃
= 𝑆 × 𝐶𝑟
4.3 Premium to Parity (%)
Parity is presented in the form of a percentage; it is the percentage the investor is willing to
spend above the market price of the share for the CB. The fact the CB will be paying coupons
to the bondholder, which in turn could be higher than the dividends paid to the
shareholders. Such a yield would increase the value of the premium, which would be of an
advantage for the firm [26]. Parity is known as the lower boundary for the CB’s speculative
value.
4.4 Investment Premium
This is used to allow the firm to know how much the investor is willing to pay for the option
to convert embedded in the CB. This is also known as the premium to the bond floor, this is
usually the percentage of the bond floor. This means the lowest price the bond could fall to
is worked out as a percentage [26].
𝑃 − 𝐵 𝐹
𝐵 𝐹
𝑃 − 𝑃𝑎
𝑃𝑎

Sanaa Khan K1306336
15
4.5 Bond Floor: 𝑩 𝑭
The bond floor is the value of the discounted cash flows of the CB. The bond floor is known
as the lower set boundary for the CB, it is usually given as a theoretical price for low share
prices of the bond. When the CB falls onto the bond floor it is looked upon as being
worthless, as it has hit the lowest price it can be if converted. As the share price increases,
the CB moves away from the bond floor and towards the conversion value [26].
The bond floor is given by:
𝐵 𝐹 = ∑ 𝐶𝑡 𝑖
𝑒(−𝑟𝑡 𝑖)
+ 𝑁𝑒(−𝑟𝑇)
𝑁 𝑐
𝑖=1
Where 𝐶𝑡𝑖 is the value of each coupon payment at the time 𝑡𝑖, r is the interest rate and 𝑁𝑐 is
the number of upcoming coupon payments [26].

Sanaa Khan K1306336
16
The graph is an illustration of the price of a CB up until year 5. The dotted line represents the
bond floor value and the parity value of the bond. As mentioned, the bond floor is the
lowest value the CB can take. The graph gives an indication as to how the bondholder would
be looking into converting this CB. The convertible’s value will be driven by the value of the
underlying shares received when the bond is converted; this is known as the parity. When
the share price is low, it is unlikely the bondholder will want to convert the bond, as they will
receive a lower amount from the issuer [26].
4.6 Price Sensitivities
The delta tells us how sensitive the price of the CB is to the economic changes to the share
price, and the exact effect it has on the CB. Delta should be closer to zero if the investor

Sanaa Khan K1306336
17
does not want the price of the CB to change. This means as the share price changes over
time it would have an impact on the face value of the bond, which is a disadvantage for the
bond issuer. Hence why it is likely they would want to ensure delta is closer to zero; meaning
if the portfolio is near zero it would not be affected by the share price changes over time.
However, if the issuer does not ensure delta is closer to zero, it would mean the value of the
convertible will be increasing and decreasing over time up until maturity.
In particular, we can see below:
∆⁡ = ⁡
𝜕𝑃
𝜕𝑆
lim
𝑆→∞
∆ = lim
𝑆→∞
𝑑𝑃
𝑑𝑆
=
𝑑
𝑑𝑆
lim
𝑆→∞
𝑃
=
𝑑
𝑑𝑆
𝑆 × 𝐶𝑟
𝑁
=
𝑑
𝑑𝑆
(𝑆) ×
𝐶𝑟
𝑁
=
𝐶𝑟
𝑁
lim
𝑆→0
∆ = 0⁡(𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔⁡𝑛𝑜⁡𝑑𝑒𝑓𝑎𝑢𝑙𝑡)

Sanaa Khan K1306336
18
The last line where delta is equal to zero can also be seen in the graph above, where the
price of the CB does not change much for low share price. If the share price is falling, then it
would mean the CB value would be shifting to the bond floor [26].
𝜕2
𝑃
𝜕𝑆2
> 0
𝜕∆
𝜕𝑆
> 0
The gamma, on the other hand, is the differential of the delta; it is looked at to ensure the
portfolio is insensitive to price movements. An increase in the gamma is usually caused
when the share price is increasing, as well as decreasing when the share price drops. The
larger the value of Γ the more likely it is affected by the change in share price [26].
The gamma tells us how often we need to rebalance the portfolio to make it ‘delta-neutral’,
i.e. how many shares need to be bought or sold in order to rebalance a portfolio of
convertible bonds. This can be very expensive for the firm to continuously rebalance, which
would mean having delta being closer to zero would be beneficial for the investor.

Sanaa Khan K1306336
19
4.7 Upper and Lower bounds
The above graph shows the price of the CB, with the upper and lower bounds. The green
dashed line which is going through the bond shows how the upper bound is formed; it is
produced with the conversion value and the share price at time t. As 𝑆𝑡 → ∞, the value of
the CB, 𝑃 ⟼ 𝐶𝑟 𝑆𝑡.
At maturity 𝑇 or before, the bondholder could exercise the bond, which would then lead to
them receiving their payoff of:
𝑃(𝑇) = max⁡( 𝑁, 𝐶𝑟 𝑆𝑡)
St
CB
CrSt
𝑁 + 𝐶𝑟 𝑆𝑡
𝑁

Sanaa Khan K1306336
20
On the other hand, the lower bound is seen as being the parity. If the price of the CB falls
below the parity then there would be a possibility for arbitrage2
profits through purchasing a
CB in the market, as well as that selling it and then replacing the borrowed shares through
conversion [30]. It has been found that it may not always be possible to short sell, however
the possibility still remains. Hence:
𝑃(𝑡) ≥ 𝐶𝑟(𝑡)𝑆(𝑡)
2
Arbitrage: The trade that makes a profit by exploiting the price differences of identical or similar
financial instruments, on different markets or in different forms. Arbitrage exists due to the result of
market inefficiencies, this provides mechanism to ensure prices do not deviate substantially from the
fair value for a long period of time.

Sanaa Khan K1306336
21
5. Pricing Methods
In this section we will be looking at pricing methods of CBs and review some of the
mathematical background needed.
5.1 Monte Carlo Simulation (MCS)
There are several pricing methods to price a CB. One method to price CBs is using the Monte
Carlo Simulation (MCS) method; this would mean using differential equations to price the
CB. The method would be suitable to find the prices of the coupon payments and the
dividend payments. Coupon payments are usually solved using a continuous method rather
than the discrete method. The Monte Carlo Simulation uses computational algorithms;
which is mainly used in three different cases: optimisation, numerical integration and from
generating draws from probability distribution. Ulam (1949) has stated how this method
uses integration, as well as probabilities to value the price of the CB. An example of the MCS
method being used would be tossing a fair coin to see if a party wins a pound coin, the
winner would gain a pound extra. In order to play the game, both players must have a pound
coin. This would mean the investor and the bond issuer are both risk seeking, as they prefer
to gamble their initial wealth.
5.2 Lattice-based method
Another way in which a bond is priced is the lattice-based method. This method consists of
producing a binomial tree, in which it would show independent and equal paths. Lattice
models are useful as they are easy to compute and they reduce computing time. Lattice

Sanaa Khan K1306336
22
methods lose accuracy when they are dealing with two or more variables; as well as losing
efficiency when discrete payments are being dealt with and early-exercise options [28].
The disadvantage of the method would be the number of nodes can increase quickly as the
number of time steps increase. [32]
5.3 Reduced Form Approach
Another method to valuate a CB is the reduced form approach, which uses the Poisson
distribution model. The reduced form simplifies the valuation of the bond; it makes an
empirical analysis of the possibility of the bonds. An advantage of this method is that it
avoids the need to determine the firms optimal call policy, also it does not require any other
source of information about the firms’ financial state. This pricing method produces a closer
fit for CB prices, as well as producing low pricing errors. This method is known to be more
appealing than other methods of valuing CBs. When valuing the CB with the reduced form
approach, the factors taken into consideration are both call and default intensities under the
risk. This specific method captures the differences between the features call and default
decisions [14].
5.4 Tsiveriotis-Fernandes (TF) Method
The method proposes to split the CB into two segments: a cash-only and equity. The cash-
only is subjected to credit risk, whereas the equity is not. The method is found to being
popular to price the CB, all because it is very simple and it holds the ability to incorporate
vital traits of CBs that have limited market data [30]. There are three methods in which this
is used to ensure the PDEs are solved efficiently, ensuring the boundary condition and

Sanaa Khan K1306336
23
discontinuities are controlled well within the calculations. The methods are: explicit method,
implicit method and the Crank-Nicolson method.
5.5 Black-Scholes (B-S) Method
B-S is a method which analyses the theory of the corporate pricing formula. In order to
derive this formula there are assumptions which the B-S model follows. The assumptions
are:
 Price of the underlying asset follows Geometric Brownian motion (GBM).
 No arbitrage opportunities.
 Unlimited short-selling.
 The risk-free interest rate is constant, and the same for borrowing and lending.
 No taxes or transaction costs.
 Underlying asset can be traded continuously and in infinitesimally small numbers of
units.
In the next section B-S will be used to model the price of the CB.
5.6 Mathematical background:
Before we proceed, we must state some mathematical definitions.
 Ω (omega) is the set of all possible outcomes, known as sample space [15].

Sanaa Khan K1306336
24
 A filtration (ℱ𝑡 ) represents the history of a process up to time t. It is a model of
information. The filtration is a collection of subsets known as ‘events’. We can assign
a probability to each of those events [15].
 ℙ is a probability measure, shows how likely an event is to happen, giving a number
𝜖[0,1] [15].
 Brownian motion (BM): A BM (also known as the Wiener process) is a stochastic
process,
𝑊 = (𝑊𝑡)t≥0 that satisfies the following:
(i) 𝑊0 ⁡= ⁡0
(ii) 𝑊 has continuous sample paths
(iii) 𝑊 has stationary increments: 𝑊𝑡 − 𝑊𝑠 ~ 𝑊𝑡−𝑠for any 0 ≤⁡𝑠 < 𝑡.
(iv) 𝑊 has independent increments
(v) 𝑊𝑡 ~ 𝑁(𝜇𝑡, 𝜎2
𝑡) for any 𝑡>0.
 Brownian motion with drift:
𝑊0 + 𝜇𝑡 + 𝜎𝑊𝑡
Where⁡𝜇𝑡 is the drift and 𝜎 is known as the diffusion coefficient or the volatility.
 Geometric Brownian motion (GBM):
A BM with drift can take negative values which is not very useful. Therefore, we use
a GBM to model the price of a stock.

Sanaa Khan K1306336
25
𝑆𝑡 = 𝑆0 𝑒
(𝜇−
𝜎2
2
)𝑡+𝜎𝑊𝑡
𝑆𝑡 denotes the stock price at time t and 𝑊𝑡 is a standard Brownian motion (SBM).
The stock price follows the GBM with stochastic differential equation [30]:
𝑑𝑆𝑡 = 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡
The drift within the equation is µ, which represents the average growth of the asset
price. The future asset price - is found from the random changes within the price;
this is seen with a random variable which is drawn from normal distribution with the
mean being zero. The σ represents the volatility in the equation, it is known as the
dimension of the standard deviation of the returns within the portfolio [24].

Sanaa Khan K1306336
26
6. Black-Scholes Model
The pricing model uses Black-Scholes (BS) to model the price of an option, an option such as
a European call option. The model follows the Geometric Brownian Motion (GBM) with a
constant drift and volatility. The method when applied to a stock option, the model itself
includes the constant price, the strike price and the time to maturity, as well as the value of
the money.
A call option is purchasing the right to buy an asset at an agreed price, on or before the
particular set date; the right within the option is only received by the investor when they pay
a premium for the option [16].
6.1. The Bond Price
The bond price is under the Black-Scholes (BS) model satisfies the BS partial differential
equation:
𝜎2
2
𝑆𝑡
2
𝑓𝑆𝑡 𝑆𝑡
+ (𝑟𝑆𝑡 − 𝐶)𝑓𝑆𝑡
− 𝑟𝑓 − 𝑓(𝑇−𝑡) + 𝑐 = 0
𝐶 denotes the coupon payments, 𝑐 being the amount of coupons paid out. 𝑟 is the interest
rate, 𝑆𝑡 is the share price and 𝜎2
is the variance return, 𝑓 in this case is: 𝑓 = (𝑆𝑡, (𝑇 − 𝑡)).
(Ingersoll 1977)

Sanaa Khan K1306336
27
However, the price of the bond can be seen as a combination of a straight bond with a call
option. This means the formula for the straight bond will be applied, as well as the formula
for the call option, if the bond is callable.
Straight bond and a call option [30]:
𝑃𝑟𝑖𝑐𝑒⁡𝑜𝑓⁡𝐶𝐵 = 𝑃𝑟𝑖𝑐𝑒⁡𝑜𝑓⁡𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡⁡𝑏𝑜𝑛𝑑 + 𝑃𝑟𝑖𝑐𝑒⁡𝑜𝑓⁡𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛⁡𝑐𝑎𝑙𝑙⁡𝑜𝑝𝑡𝑖𝑜𝑛
𝑋(𝑡) = 𝑁 ∑ 𝐶𝑒−𝑟(𝑡 𝑖−𝑡)
+ 𝑁𝑒−𝑟(𝑇−𝑡)
𝑛
𝑖=1
Where 𝑡𝑖 represents the dates at which the coupons are paid.
Price of a European call option [30]:
𝑉(𝑡) = 𝑆𝑡Φ(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)
Φ(𝑑2)
Where 𝑑1 and 𝑑2 are [30]:
𝑑1 =
log (
𝑆𝑡
𝐾
) + (𝑟 +
𝜎2
2
)(𝑇 − 𝑡)
𝜎√ 𝑇 − 𝑡

Sanaa Khan K1306336
28
𝑑2 = 𝑑1 − 𝜎√𝑇 − 𝑡
The value of Φ(𝑑1) and Φ(𝑑2) is found to be the cumulative probability distribution
function for a standard normal distribution.
This then leads to:
𝑃(𝑡) = 𝑋(𝑡) + 𝑉(𝑡)
This is finding the final value of the price of the CB, finding the sum of the two components.
However, splitting the two components relies on restrictive assumptions, an example being
the embedded options. Splitting the components mean not being able to call the bond, nor
being able to sell it back to the issuer; which are the features available within a CB. However,
these cannot be taken into consideration with the BS equation above [30]. As a convertible
usually is American styled, it becomes hard to value it, in terms of the closed-form approach
of the BS model. The closed-form approach is computing the value of the option; the
method gives the issuer an idea of the pricing and the behavior of the CB. The method
becomes complicated to use when continuous time intervals are used within the option; it
becomes difficult to find the price if early exercise is available within the option. Hence, for
CBs it is not appropriate, unless it is European in nature [27].
The payoff for a call option is: 𝑚𝑎𝑥⁡( 𝑆 𝑇 − 𝐾, 0), the bondholder has two options from which
they will receive one, (i) if 𝑆 𝑇 − 𝐾 is negative then the payoff value is 0 and (ii) if 𝑆 𝑇 − 𝐾 is
positive then the payoff value is an integer that the bondholder will receive.

Sanaa Khan K1306336
29
The fair price of a CB is defined using the replicating strategy, using units of stock and cash;
this continues to follow the BS model of valuation.
6.2. Example
Initial Stock (S0) 300
Time (T) 5
Strike Price (K) 350
Risk-free rate (r ) 0.02
Volatility 0.1
Face Value (N) 500
Coupon Payments (C ) 0
d1 0.18305321
d2 -0.04055359
Call option V(t) 18.56223972
Bond Value Bf 0.455940983
Convertible Bond (CB) 19.018180713
Table 1 shows the current data for a CB and the price of it when the components have been
split into a call option and a straight bond. Using these parameters two graphs will be
plotted, (i) which will contain the increase of the strike price, whilst the other parameters
remain the same and (ii) where the interest rate will be increasing, again whilst the rest of
3
Table 1

Sanaa Khan K1306336
30
the parameters remain the same. This will help to show the impact of changing a parameter
and how this affects the price of the CB.
4
The graph shows the impact of the change in the conversion price, it shows how the price of
the CB price has decreased as the value of 𝐶 𝑃 has increased. The other parameters remain
the same, however it is seen how the value of the CB changes with the change in a
parameter value. As the conversion price increases, the value of the call option is out-of-the-
money, which means the payoff would be equal to zero as the conversion price continues to
increase. If the conversion price is high, the bondholder would be able to purchase shares in
the market, more shares than which they would receive at conversion. In this case, the
bondholder would not be exercising the CB and would see benefit in buying the shares in the
market.
4
Graph 1: change in conversion price
0
50
100
150
200
250
150 200 250 300 350
Price of CB with increasing
conversion price
P(t)
Conversion Price (𝐶 𝑃)
ConvertibleBondPrice

Sanaa Khan K1306336
31
5
The graph shows the impact of the change in the interest rate on the price of a CB. The
graphs shows as the rate of interest increases the CB is worth more, the same can be said for
the call option; an increase in the interest rate, increases the value of the option and the CB.
6.3. Margrabe Formula
The Margrabe formula generalises the BS pricing model to price options, which gives the
holder of the option the right to exchange but not the obligation to exchange ‘S’ units of one
asset into ‘P’ units of another [25]. In this case, a CB can be viewed as a risky straight bond
and the option to exchange the straight bond for a specific amount of shares. The Margrabe
model assumes that the assets follow the GBM with the correlation 𝜌 [30]. In this case,
5
Graph 2: change in interest rate
0
10
20
30
40
50
60
0.02 0.03 0.04 0.05 0.06
Price of CB with increasing interest
rate
P(t)
Risk-free Interest rate (𝑟)

Sanaa Khan K1306336
32
however, there are contradictions to applying the GBM to the CBs; this is seen with the
results, as the results we obtain are sensible.
The equation for the replicating portfolio (exchange option) is shown by [25]:
𝐸(𝑡) = 𝑄1 𝑆1 𝑒((𝑎1−𝑟)(𝑇−𝑡))
Φ(𝑑1) − 𝑄2 𝑆2 𝑒((𝑎2−𝑟)(𝑇−𝑡))
Φ(𝑑2)
Where [25]:
𝑑1,2 =⁡
ln (
𝑆1
𝑆2
) + (𝑎1 − 𝑎2 ±
𝜎̂2
2
)(𝑇 − 𝑡)
𝜎̂√ 𝑇 − 𝑡
𝜎̂ = √𝜎1
2
𝜎2
2
− 2𝜌𝜎1 𝜎2
Where 𝑎1 = 𝑎2 = 𝑟, 𝑆1, 𝑆2 are assets which are chosen to be exchanged and 𝑄1⁡and 𝑄2 are
the quantities of the assets. There are no dividend payments in this case; therefore, early
exercise of the CB will not be optimal for the bondholder [25]. The definition of Φ(𝑑1) and
Φ(𝑑2) can be found in section 6.1.

Sanaa Khan K1306336
33
An exchange option is seen to be European in nature, which means it cannot be called by the
issuers and there are no coupon payments made [30]. In this case, under the assumption of
the Margrabe model, 𝑆1⁡can be seen as our share price 𝑆𝑡 and 𝑄1 following to be the
conversion ratio 𝐶𝑟, 𝑆2 is the price of the bond 𝑃(𝑡). Since there is only one unit of the bond
this means 𝑄2 = 1. The price of the bond 𝑃(𝑡), at time 𝑡 = 0 is, 𝑃(0) = 𝑁𝑒−𝛿𝑇
. Where 𝛿 is
the continuous compounded yield rate.
The replicating portfolio, as seen as above, consists of 𝐶𝑟 𝑒((𝑎1−𝑟)(𝑇−𝑡))
Φ(𝑑1) amount of
shares and 𝑒((𝑎2−𝑟)(𝑇−𝑡))
Φ(𝑑2) of loaned cash. When there is a change in the share price,
i.e: 𝑆𝑡 → ∞ then, Φ(𝑑1); Φ(𝑑2) → 1 this then leaves the replicating strategy being a long
position in share value: 𝑆𝑡 𝐶𝑟 𝑒((𝑎1−𝑟)(𝑇−𝑡))
and a short position with the cash amount:
𝑁𝑒−𝛿𝑇+((𝑎1−𝑟)(𝑇−𝑡))
. This gets balanced out due to the risky long position, which then leads
to: as 𝑆𝑡 → ∞, 𝑃(𝑡) = 𝑆𝑡 𝐶𝑟 𝑒((𝑎1−𝑟)(𝑇−𝑡))
[30].
Using the value of 𝑑𝑆𝑡 (section 5.6), following the BS model the value of the call option
becomes [30]:
𝑉(𝑡) = 𝑆𝑡Φ(𝑑1) − 𝐾𝑡 𝑒−𝑟(𝑇−𝑡)
Φ(𝑑2)
Where in this case 𝑑1 and 𝑑2 are equal to [30]:
𝑑1,2 =
ln (
𝑆𝑡
𝐾) + (𝑟 ±
𝜎2
2 )(𝑇 − 𝑡)
𝜎√ 𝑇 − 𝑡
A more detailed discussion of the margrabe formula is left for future work.

Sanaa Khan K1306336
34
7. Conclusion
Convertible bonds (CB) are interesting as they combine two financial instruments. We see
that CBs can be modelled as a zero-sum game between the bondholder and the bond issuer.
This paper focuses on the pricing of a CB, going through stages of the bond before finally
breaking down the price of the bond, as well as discussing the methods used to price the
bond. An example of these is Black-Scholes model, it is commonly used to help price the
bond, however we also find that using the lattice-based method also works well. Under the
Black-Scholes model, we are able to split the components of the CB into a straight bond and
a call option.
The interesting factor is that the issuer and the bondholder are able to change the features
within the bond according to their needs, which means both parties have a fair advantage. A
feature such as the premium to pay the issuer would be discussed and can be changed
before signing for the bond. This means the CB are attractive for both the issuer and the
bondholder, as it helps the issuer with financing their needs and gives the bondholder the
right to convert when they feel it is beneficial for them.
The paper also looks at an example of a CB for a specific set of parameters. We see how that
affects the overall price of the CB and whether it increases or decreases with changes in
interest rate and conversion price.
CBs offer the investor a greater right than the issuer of the bond – the right for which they
have paid for, thus it is acceptable. However, the instruments used within the contract must
be carefully understood by both parties in order to be able to make the investment.

Sanaa Khan K1306336
35
8. References:
[1] – Amiram. D, Kalay. A, Kalay. A and Ozel. B. N (2014) ‘Capital Market Frictions and Bond Coupon
Choice’. Ph. D thesis, Columbia Business School [Online]. Available
at:https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/11551/AKKO%2012-8-14.pdf
(Accessed 25
th
December 2015).
[2] - Ammann. M, Kind. A and Wilde. C (2008) 'Simulation-based Pricing of Convertible Bonds', Journal
of Empirical Finance, 15(2), pp. 310-331 [Online]. Available
at:http://www.sciencedirect.com/science/article/pii/S0927539807000618 (Accessed: 9th October
2015).
[3] - Batten. A. J, Khaw. H-L. K and Young. R. M (2014) 'Pricing Convertible Bonds', Journal of Economic
Survey, 28(5), pp. 775-803 [Online]. Available at:
http://onlinelibrary.wiley.com/doi/10.1111/joes.12016/epdf (Accessed: 13th October 2015).
[4] - Beveridge. C and Joshi. S. M (2010) 'Monte Carlo Bounds for Game Options Including Convertible
Bonds', Management Science, 57(5), pp. 960 - 974 [Online]. Available
at:http://fbe.unimelb.edu.au/__data/assets/pdf_file/0010/806347/207.pdf (Accessed: 26th January
2016).
[5] – Brennan. J. M and Schwartz. S. E (1980) ‘Analyzing Convertible Bonds’, Journal of Financial and
Quantitative Analysis, 15(4), pp. 907-929 [Online]. Available at:
http://journals.cambridge.org/download.php?file=%2FJFQ%2FJFQ15_04%2FS0022109000013855a.pd
f&code=eee71f7a48bf9f0cd563057d88374391 (Accessed 25
th
December 2015).
[6] – Chan. H. W. A and Chen. N (2004) ‘Convertible Bond Pricing: Renegotiable Covenants, Seasoning
and Convergence’. Ph. D thesis, Swedish School of Economics [Online]. Available at:
http://merage.uci.edu/resources/documents/convertible%20bonds.pdf (Accessed: 20
th
December
2015).

Sanaa Khan K1306336
36
[7]- Chen. N, Dai. M and Wan. X (2013) 'A Nonzero-Sum Game Approach to Convertible Bonds: Tax
Benefits, Bankruptcy Cost, and Early/Late Calls', Mathematical Finance,23(1), pp. 57-93 [Online].
Available at: http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9965.2011.00488.x/epdf (Accessed:
8th October 2010).
[8] - De Souza, A. (2010) ‘How would bondholders vote?’, MSc thesis, New York University [Online].
Available at: http://ejournal.narotama.ac.id/files/How%20would%20bondholders%20vote.pdf
(Accessed 27
th
December 2015).
[9] - Garikaparthi. M (2014) 'NASH EQUILIBRIUM – A STRATEGY OF GAMES THEORY',International
Journal of DEVELOPMENT RESEARCH, 4(7), pp. 1316-1318 [Online]. Available
at: http://www.journalijdr.com/sites/default/files/1679_0.pdf (Accessed: 24th January 2016).
[10] - Gillet. R and Bruslerie. L. D. H (2010) 'The Consequences of Issuing Convertible Bonds: Dilution
and/or Financial Restructuring?', European Financial Management,16(4), pp. 552-584 [Online].
Available at: https://halshs.archives-ouvertes.fr/halshs-00674248/document (Accessed: 2nd October
2015).
[11] – Grimwood. R and Hodges. S (2002) ‘The Valuation of Convertible Bonds: A Study of Alternative
Pricing Models’, PHD thesis, 1(1), pp. 1-82 [Online]. Available at:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.202.4571&rep=rep1&type=pdf (Accessed
16th October 2015).
[12] - Ibrahim, Y and Hwei, L. K (2010) 'Firm Characteristics and the Choice between Straight Debt and
Convertible Debt among Malaysian Listed Companies', International Journal of Business and
Management, 5(11), pp. 74-83 [Online]. Available
at:http://www.ccsenet.org/journal/index.php/ijbm/article/view/8056/6083 (Accessed: 9th October
2015).
[13] - Ingersoll. E. J (1977) 'A Contingent-Claims Valuation of Convertible Securities', Journal of
Financial Economics, 4(3), pp. 289–321 [Online]. Available at:

Sanaa Khan K1306336
37
http://faculty.som.yale.edu/jonathaningersoll/downloads/1977_ConvertibleSecurities.pdf (Accessed:
13th October 2015).
[14] - Jarrow. R, Li. H, Liu. S and Wu. C (2010) 'Reduced-Form Valuation of Callable Corporate Bonds:
Theory and Evidence', Journal of Financial Economics, 95(2), pp. 227–248 [Online]. Available
at: http://ac.els-cdn.com/S0304405X09002074/1-s2.0-S0304405X09002074-main.pdf?_tid=bf4536b2-
b5ac-11e5-95d4-00000aab0f02&acdnat=1452219105_30f70d7f70a288d27b7bfb0d720c0a03
(Accessed: 6th January 2016).
[15] - Karatzas. I and E. Shreve. S (1991) Brownian Motion and Stochastic Calculus, 2 edn., United
States of America: Springer Science and Business Media.
[16] - Kolb, R.W. (1995) Understanding options, 2nd edn.,: Wiley Finance.
[17] - Koziol. C (2004) Valuation of Convertible Bonds when Investors act Strategically, 1 edn.,
Frankfurt: Springer Science and Media Business.
[18] - Kwok. K. Y (2014) 'Game Option Models of Convertible Bonds: Determinants of Call
Policies', Journal of Financial Engineering, 1(4), pp. 1-19 [Online]. Available
at:https://www.math.ust.hk/~maykwok/piblications/Kwok_JFE_2014.pdf (Accessed: 13th October
2015).
[19] - Lau. W. K and Kwok. K. Y (2004) 'Anatomy of Option Features in Convertible Bonds', The Journal
of Futures Markets, 24(6), pp. 513–532 [Online]. Available at:
http://www.math.ust.hk/~maykwok/piblications/Articles/anatomy_jour.pdf (Accessed: 1st November
2015).
[20] – Liu. J and Switzer. N. L (2013) 'Convertible Bond Issuance, Risk, and Firm Financial Policy: A New
Approach', INTERNATIONAL JOURNAL OF BUSINESS, 18(1), pp. 1-18 [Online]. Available at:
http://search.proquest.com.ezproxy.kingston.ac.uk/docview/1321093946/fulltextPDF?accountid=145
57 (Accessed: 16th October 2015).

Sanaa Khan K1306336
38
[21] - Loncarski. I, Horst. T. J and Veld. C (2005) 'Why do Companies Issue Convertible Bonds? A
Review of Theory and Empirical Evidence', Advances in Corporate Finance and Asset Pricing, 10(1), pp.
1-34 [Online]. Available
at:http://www.researchgate.net/publication/228319183_Why_do_Companies_Issue_Convertible_Bon
ds_A_Review_of_Theory_and_Empirical_Evidence(Accessed: 2nd October 2015).
[22] – Lvov. D, Yigitbasioglu. B. A and El Bachir. N (2004) ‘Pricing Convertible Bonds By Simulation’. Ph.
D thesis, University of Reading. Available [Online]: http://www.icmacentre.ac.uk/files/discussion-
papers/DP2004-15.pdf (Accessed 13th October 2015)
[23] – Merton. C. R (1976) 'Option Pricing when Underlying Stock returns are Discontinuous', Journal
of Financial Economics , 1(3), pp. 125-144 [Online]. Available
at:http://www.people.hbs.edu/rmerton/optionpricingwhenunderlingstock.pdf (Accessed: 16th March
2016).
[24] – Partanen. D. B and Jarnberg. E (2014) ‘Convertible Bonds: A Qualitative and Numerical
Analysis’. BSc thesis, KTH Royal Institute of Technology [Online]. Available at: http://www.diva-
portal.org/smash/get/diva2:729641/FULLTEXT01.pdf (Accessed 12th January 2016)
[25] – Poulsen. R (2010) ‘The Margrabe Formula’, Encyclopedia of Quantitative Finance, 1(1), pp.
1118-1120 [Online]. Available at:
http://webcache.googleusercontent.com/search?q=cache:sDQcMZF2D8EJ:www.math.ku.dk/~rolf/EQ
F_Margrabe.pdf+&cd=1&hl=en&ct=clnk&gl=uk (Accessed: 16
th
March 2016)
[26] - Spiegeleer De. J, Schoutens. W and Hulle. V. C (2014) The Handbook of Hybrid Securities:
Convertible Bonds, Coco Bonds, and Bail-In, UK: CPI Group.
[27] – Sundaram. K. R and Das. R. S (2011), Derivatives: Principles and Practice, 1 edn., McGraw-Hill
Irwin, New York.
[28] - Ulam. S (1949) 'The Monte Carlo Method', Journal of the American Statistical Association,
44(247), pp. 335-341 [Online]. Available at: http://www.amstat.org/misc/TheMonteCarloMethod.pdf
(Accessed: 13th October 2015).

Sanaa Khan K1306336
39
[29] -Yan. H, Yi. F, Yang. Z and Liang. G (2015) 'Dynkin Game of Convertible Bonds and their Optimal
Strategy', Journal of Mathematical Analysis and Applications, 426(1), pp. 64-88 [Online]. Available at:
http://www.sciencedirect.com/science/article/pii/S0022247X15000591 (Accessed: 23rd October
2015).
[30] – Zadikov. A (2010) ‘Methods of Pricing Convertible Bonds’. MSc thesis, University of Cape Town
[Online]. Available at:
http://www.mth.uct.ac.za/academics/postgrad/graduatethesis/MSc_Ariel_Zadikov.pdf (Accessed
22nd December 2015).
[31] - Zhang. G- W and Liao. K- P (2014) 'Pricing Convertible Bonds with Credit Risk under Regime
Switching and Numerical Solutions', Mathematical Problems in Engineering, 2014(1), pp. 1-13
[Online]. Available at: http://www.hindawi.com/journals/mpe/2014/381943/ (Accessed: 13th
October 2014).
[32] - Zhao. L. J and Liu. H. R (2013) 'A lattice Method for Option Pricing with two Underlying Assets in
the Regime-Switching Model', Journal of Computational and Applied Mathematics, 250(1), pp. 96-106
[Online]. Available at: http://ac.els-cdn.com/S0377042713000812/1-s2.0-S0377042713000812-
main.pdf?_tid=d95d5106-fc33-11e5-ad95-
00000aab0f01&acdnat=1459973712_da96d1eeb35e618458f9c49254b40052 (Accessed: 31st March
2016).

Valuation of Convertible Bonds pdf

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (6)

Similar to Valuation of Convertible Bonds pdf

Similar to Valuation of Convertible Bonds pdf (20)

Valuation of Convertible Bonds pdf